Thesis

The Dirac–Higgs bundle is a
hyperholomorphic bundle on the hyperkähler moduli space
of Higgs bundles. Its construction parallels that of a
Nahm transform for instantons, but just for Higgs bundles, by using a Dirac operator coupled to a Higgs
bundle. The thesis disucsses many different aspects of the
Dirac–Higgs bundle, among other things we consider the bundle
for parabolic Higgs bundles and use it to construct
doubly-periodic instantons. The holomorphic description of a
Nahm transform is typically called a Fourier–Mukai
transform. We also discuss the Dirac–Higgs bundle from the
Fourier–Mukai perspective.

Outreach

I've had the opportunity to interview several high profile mathematicians about their views upon mathematics, and their entry into maths. The results can be viewed on the page in the link or by clicking on the names below.

This document is an assessment for a course on derived categories of coherent sheaves, in the fall of 2011, at Oxford University. The document is a review of aspects of derived categories of coherent sheaves. We review the definitions of derived categories and of coherent sheaves on a smooth projective variety. Following this we will discuss properties such as Serre duality and applications of Serre duality. Lastly we will discuss the Bondal–Orlov Reconstruction theorem.

This document is an assessment for a course on symplectic geometry, in the fall of 2011, at Oxford University. The document is a review of aspects of mirror symmetry with special attention to the predictions about the number of rational curves on a quintic threefold. The calculations leading to the predictions are reviewed, and afterwards Gromov–Witten invariants for quintic three-folds are defined and calculations are discussed.

This is my progress report. It is written as the conclusion of my first 3 years as a PhD-student. It contains a chapter about complex geometry, quantization, the Hitchin connection, moduli spaces of flat connections, abelian varieties, Berezin-Topelitz deformation quantization, geometric quantization of abelian varieties.

We calculate the moduli space of flat SU(2) connections on several surfaces. The surfaces in question are the torus with zero and one puncture, and the sphere with one, two, three and four punctures. Included is also a chapter with serveral nice facts about SU(2) and its lie algebra. Besides all this is a chapter dedicated to the SU(2) representation variety, discussing its topology and the dimension in the case of a surface group.

These notes were written for a talk I gave at University of California, Berkeley in November 2010. The talk was about a method of converting link invariants to 3-manifold invariants. The example in mind are the BHMV-invariants derived from skein theory and the Kauffman bracket.

This is a note on the
mathematics behind Google.The first part of the note is
mainly about the philosophy behind Google, and a small toy
example of how it works. The last part is the mathematical
part, where it is proven using basic linear algebra, how
Google works. The paper is written in Danish.

In this note we list a lot of
facts about classical Lie groups. The focus will mainly be
on dimension, compactness, connectedness and relationships
between the classical Lie groups. Some basic principal
bundles of the classical Lie groups will also be
listed.

In this note we investigate the notion of a group action on a topological space. First of all what it is, but also see that almost all such spaces have the structure of a quotient space. The spaces which have a quotient space structure are called homogeneous spaces. Last but not least we will be dealing with some examples of homogeneous spaces.

This note is part of the evaluation in the course Unitary group representations.

I have written a note on two representations of the hyperbolic plane: The disc model and the Upper Half-plane model. In particular we will look at the geodesics in the two representations, and especially discuss the distance between two arbitrary points in the disc model. Last but not least it will be shown that the Gaussian curvature of the hyperbolic plane is constantly -1, by computing the Gaussian curvature in the Upper Half-plane model.

My B.Sc. in mathematics was completed with this assignment on semigroups of contraction operators. I prove the classical theorem by Hille and Yoshida, and another classical theorem of Stones. The paper is written in Danish.